Optimal. Leaf size=112 \[ -\frac {2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4 (d+e x)^{3/2}}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac {2 c (3 B d-A e)}{e^4 \sqrt {d+e x}}+\frac {2 B c \sqrt {d+e x}}{e^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {772} \begin {gather*} -\frac {2 \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4 (d+e x)^{3/2}}+\frac {2 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac {2 c (3 B d-A e)}{e^4 \sqrt {d+e x}}+\frac {2 B c \sqrt {d+e x}}{e^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 772
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )}{e^3 (d+e x)^{7/2}}+\frac {3 B c d^2-2 A c d e+a B e^2}{e^3 (d+e x)^{5/2}}+\frac {c (-3 B d+A e)}{e^3 (d+e x)^{3/2}}+\frac {B c}{e^3 \sqrt {d+e x}}\right ) \, dx\\ &=\frac {2 (B d-A e) \left (c d^2+a e^2\right )}{5 e^4 (d+e x)^{5/2}}-\frac {2 \left (3 B c d^2-2 A c d e+a B e^2\right )}{3 e^4 (d+e x)^{3/2}}+\frac {2 c (3 B d-A e)}{e^4 \sqrt {d+e x}}+\frac {2 B c \sqrt {d+e x}}{e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 95, normalized size = 0.85 \begin {gather*} -\frac {2 \left (3 a A e^3+a B e^2 (2 d+5 e x)+A c e \left (8 d^2+20 d e x+15 e^2 x^2\right )-3 B c \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )}{15 e^4 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.07, size = 117, normalized size = 1.04 \begin {gather*} \frac {2 \left (-3 a A e^3-5 a B e^2 (d+e x)+3 a B d e^2-3 A c d^2 e+10 A c d e (d+e x)-15 A c e (d+e x)^2+3 B c d^3-15 B c d^2 (d+e x)+45 B c d (d+e x)^2+15 B c (d+e x)^3\right )}{15 e^4 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 133, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (15 \, B c e^{3} x^{3} + 48 \, B c d^{3} - 8 \, A c d^{2} e - 2 \, B a d e^{2} - 3 \, A a e^{3} + 15 \, {\left (6 \, B c d e^{2} - A c e^{3}\right )} x^{2} + 5 \, {\left (24 \, B c d^{2} e - 4 \, A c d e^{2} - B a e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 122, normalized size = 1.09 \begin {gather*} 2 \, \sqrt {x e + d} B c e^{\left (-4\right )} + \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} B c d - 15 \, {\left (x e + d\right )} B c d^{2} + 3 \, B c d^{3} - 15 \, {\left (x e + d\right )}^{2} A c e + 10 \, {\left (x e + d\right )} A c d e - 3 \, A c d^{2} e - 5 \, {\left (x e + d\right )} B a e^{2} + 3 \, B a d e^{2} - 3 \, A a e^{3}\right )} e^{\left (-4\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 101, normalized size = 0.90 \begin {gather*} -\frac {2 \left (-15 B c \,x^{3} e^{3}+15 A c \,e^{3} x^{2}-90 B c d \,e^{2} x^{2}+20 A c d \,e^{2} x +5 B a \,e^{3} x -120 B c \,d^{2} e x +3 a A \,e^{3}+8 A c \,d^{2} e +2 a B d \,e^{2}-48 B c \,d^{3}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 109, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (\frac {15 \, \sqrt {e x + d} B c}{e^{3}} + \frac {3 \, B c d^{3} - 3 \, A c d^{2} e + 3 \, B a d e^{2} - 3 \, A a e^{3} + 15 \, {\left (3 \, B c d - A c e\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{3}}\right )}}{15 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.74, size = 100, normalized size = 0.89 \begin {gather*} -\frac {2\,\left (-48\,B\,c\,d^3-120\,B\,c\,d^2\,e\,x+8\,A\,c\,d^2\,e-90\,B\,c\,d\,e^2\,x^2+20\,A\,c\,d\,e^2\,x+2\,B\,a\,d\,e^2-15\,B\,c\,e^3\,x^3+15\,A\,c\,e^3\,x^2+5\,B\,a\,e^3\,x+3\,A\,a\,e^3\right )}{15\,e^4\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.22, size = 653, normalized size = 5.83 \begin {gather*} \begin {cases} - \frac {6 A a e^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {16 A c d^{2} e}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {40 A c d e^{2} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {30 A c e^{3} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {4 B a d e^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {10 B a e^{3} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {96 B c d^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {240 B c d^{2} e x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {180 B c d e^{2} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {30 B c e^{3} x^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a x + \frac {A c x^{3}}{3} + \frac {B a x^{2}}{2} + \frac {B c x^{4}}{4}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________